Theorem in3an

Description: The virtual deduction introduction rule converting the second conjunct of the third virtual hypothesis into the antecedent of the conclusion. exp4a is the non-virtual deduction form of in3an . (Contributed by Alan Sare, 25-Jun-2012) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	in3an.1	⊢ (    𝜑    ,    𝜓    ,    ( 𝜒 ∧ 𝜃 )    ▶    𝜏    )
Assertion	in3an	⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    ( 𝜃 → 𝜏 )    )

Proof

Step	Hyp	Ref	Expression
1		in3an.1	⊢ (    𝜑    ,    𝜓    ,    ( 𝜒 ∧ 𝜃 )    ▶    𝜏    )
2	1	dfvd3i	⊢ ( 𝜑 → ( 𝜓 → ( ( 𝜒 ∧ 𝜃 ) → 𝜏 ) ) )
3	2	exp4a	⊢ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜃 → 𝜏 ) ) ) )
4	3	dfvd3ir	⊢ (    𝜑    ,    𝜓    ,    𝜒    ▶    ( 𝜃 → 𝜏 )    )